We have .

### 1.a

Let’s first prove that is symmetric:

.

So we have by uniqueness of the inverse.

Proving that is a projection matrix:

.

### 1.b

### 1.c

### 1.d

### 1.e

#### (i)

#### (ii)

We look for with and .

Now we have different cases:

##### First column (j = 1)

So the first column has 1 in the first row, and 0 in all the others.

##### First row (i = 1)

So the first row has 1 in the first column and in the others.

##### Diagonal (i = j)

So the diagonal is all 1.

##### General values (i != j) (i != 1) (j != 1)

##### General

Checking:

#### (iii)

#### (iv)

We want to analyze . Let’s try to get the elements of more formally. We know that

,

so we will have

.

Analyzing the product:

.

Let’s analyze the cases separately:

##### i = 1 and j = 1

##### i = 1 and j != 1

##### i != 1 and j = 1

(same reason)

##### i != 1 and j != 1

.

If we use

,

then we get

,

finishing the proof.